The density topology on **R** consists of
all measurable
subsets A of **R** such that, for every x in A,
x is a density point of A. It is a completely regular
refinement of the natural topology. A function
f:**R**-->**R** is *density continuous* if and
only if it is continuous as a selfmap of **R** equipped
with the density topology.

Throughout this paper we are concerned with the relationship between
density continuity and differentiability. In the process, we discuss
the fact that any closed set can be made into the zero set of a
C^{infty} density continuous function, and we show that there is a
nowhere approximately differentiable density continuous and continuous
function. This example answers a problem posed by Ostaszewski.

**Last modified August 23, 1999.**